Notes on Blackbody radiation

 Hi folks sorry  I have been silent for a while anyway I have beeen busy for about a year writing up some notes on Blackbody radiation. This is a fairly straightforward run through of the basic ideas of Statistical Mechanics and how Planck managed to finally obtain the correct formula by making the guess that electromagneitic energy is quantised. For Planck this was just a ruse and he thought little about it. However it took Einstein to take the idea seriously and he applied it to calculate the specific heat of solids and almost got the correct low temperature behaviour. It was left to Debye to improve on Einstein's model by performing a numerical integration over all frquencies. So Debye is a better physicist than Einstein 😅😅Well at least in that respect. 

I must admit I have had a love hate relationship with Statistical Physics over the years we had a pretty bad lecturer a certain Dr Jones at Exeter university who was (is) a really clever guy but his lectures bore no real resemblance to conventional statistical physics texts and so I and a friend who I was studying with at the time were left scratching our heads and wondering what the subject was all about. With these set of notes I have finally laid the ghost of that experience to reat and now feel I have enough background in statistical physics to understand its applications in Astro-physics especially the physics of the early universe and the calculation of the Helium abundance in the universe by Peebles and other people. 

Of course everyone knows that the Cosmic Background radiaton is a black body and I produce a graph comparing Planck's formula with the FIRAS data obtained from the COBE satellite. I also show how to do the Numerical Integration to get Debye's prediction of the specific heat of silver at low temperatures. 

There are some pretty interesting integrals which invove the Product of the Riemann Zeta function and the Gamma Function. and I show how these are derived unlike most books which just state the formula. As this formula is ubiquitous in Astro-Physical applications then I have enclosed it in a red box at the end of a fairly long appendix at the end. 

Anyway I hope you find the notes useful I now have all the building blocks in place to understand the Peebles calculation and over the next year or two I hope to finally finish this culminating in a code which traces the abundance of the Light elements during the first three minutes after the biig bang 

Here is a link to the file 

https://drive.google.com/file/d/11tpWAP4vh1ywl9ITv8opKCCsiJBws2oP/view?usp=sharing

Cambridge NST Maths year 1 2019 Paper 2 solutions

 Here is the second in a sequence of my solutions to the Cambridge NST maths papers, This one is for the second of the papers set for the first year students in 2019 the last year before COVID. You can access it here 

https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing

I have previously published my solutions to the first of the papers and for convenience I post the link again here for those who missed it first time around

https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing

So there you have it a complete set of answers to the first year papers for NST students at Cambridge for 2019. Most of them will have sat there finals last summer and I hope they did well. 

I apologise in advance for any typos spelling errors etc. A review of the paper follows 

Again just like the first paper this was quite challenging and unfortunately I was unable to answer a question on Parseval's theorem properly. On the whole though I think I got the questions out, but I doubt if I could do well under exam conditions. 

Anyway just like the first paper there were 10 short questions which were relatively straightforward once you decoded what the examiner was getting at. The questions included solving a first order differential equation by the integrating factor method. Deriving a recurrence relation for an Integral (something they love testing people on). Calculating the stationary values of a function f(x,y). A slightly confusing question on pronability I really need more practice at questions involviing conditional probability. A volume integral and a surface integral. All this should take no longer than 30 mins but I suspect I would take a lot longer. Again there is no time to think and in the rush you would probably end up making silly mistakes,

The core of the paper is 10 questions of which answers to five must be submitted. The last two questions are reserved for those students deemed clever enough to do some advanced topics although I didn't think they were particularly difficult. 

Anyway here are the questons 

Question 11 was a geometric one involving the equation of a plane and finding the volume of the parallipped enclosed by 3 vectors. This was relatively straightforward once I had reminded myself of the vector equation of a plane. But there were quite a few parts. Although the question asked for a few diagrams I had the luxury of using MATLAB to draw the relevant pictures/ Not very exciting I must admit

Question 12 . Involved fiinding the stationary points of a function in f(x,y) and drawing a contour plot showing the function and the gradient. This was tedious but relatively straightforward and again I was able to use MATLAB to draw some pretty pictures. 

Question 13 Involved calculating the line integral of a vector function F for various paths and also finding a function satisfying curl F = 0. (a conservative function). I hadn't done a question like this for ages so I had to remind myself of how you go about calculating such things but it was relatively straightforward although finding the conservative function involved a little guesswork so not very satisfactory.

Question 14  Involved some questions on probability density functions. and evaluating their products and change of  variables. It was ok but not a very exciting topic. 

Question 15  This was a set of questions on solving second order differential equations with constant coefficients my favourite topic at this level. The first question was a homogeneous equation with boundary conditions and relatively straightforward to solve. The second part was an inhomogenous equation and whilst finding the complementary function was relatively straightforward. In order to find the particular integral you had use a function of the form  x^n f(x) and increase n until you found one that worked. This took a couple of goes and so would have been quite time consuming under exam conditions how nice of them 😊. The last part involved solving two differential equations simultaneously some what surprisingly they don't seem to teach how to solve such systems using matrices and their eigenvectors unlike the open university courses MST210 or MST224 so this is one occasion wihere the open university is better than Cambridge. Anyway compared to the tedium of the last few questions this was a delight to do. 

Question 16   This question was all about calculating various surface integrals and the flux of a field through a surface. It got a bit fiddly but again was relatively straight forward. Another boring topic though I much prefer solving differential equations 

Question 17 This was a boring question on matrices again pretty straighrforward but you have to know the definitions and again there were so many parts to the question. Give me calculus questions any day

Question 18  This was a question on Fourier series you had to find the Fourier series for cosh(x) then differentiate to get the Fourier series for sinh(x). For this topic you really need to be on top of integration by parts. The last part of the question then asked you to use Parsevals theorem to show that the integral of (cosh(x)-sinh(x))^2 over the interval was sinh(2) . I tried this a couple of times but I couldn't get the expansions to cancel out to leave sinh(2) so unfortunately I was unable to complete the paper properly. However if you integrate the function directly it comes out relatively straightforwardly. 

So the two questions for the so called advanced students were as follows 

Question 19 was on Lagrangian multipliers and you had to find the optimum volume of a cylinder the optimum volume of a cone inscrbed in a sphere and then prove that the Arithmetic mean is >= to the geometric mean. I confess to nor really understanding this topic although I can go through the motions and I find it difficult to tell whether I am finding a minimum or a maximum. For the cone inscribed inside a sphere I found it easier to just finding the maximum volume directly. I'll let you solve this question using Lagrangian Multipliers fot your self. 

Question 20  A relatively straightforward question on solving partial differential equations using separation of variables. Two first order ones and a question on the diffusion equaton. This is really a warm up for what comes next year so you aren't asked to solve the differential equations in spherical or cylindrical coordinate systems or use Lagrange Polynomials or Bessel functions or any of the other exotic functions out there. So a bit boring really 

Overall conclusion is that this was an exercise worth doing to remind myself and extend my mathematical knowledge a bit. I think on the whole I preferred the first paper as it seemed to cover slightly more interesting topics. Apart from the two quesitions on differential equations this paper could be described as worthy but dull. 

The second year papers for this year beckon next and I hope that I find them a bit more interesting than this one. Hopefully I can finish them by June next year

I would urge you to have a go for yourself and I hope you find these solutions useful 

Thermodynamics

 The laws of thermodynamics are fundamental for understanding the  structure of matter and the transfer of heat. Remarkably they stand by themselves and have no need of any microscopic underpinning This point is often missed in treatments of thermodynamics which quickly move onto statistical physics and don't encourage physicists to develop their powers of thermodynamic reasoning. 

The best account of thermodynamics I know of is given in Longairs book 

https://www.amazon.co.uk/Theoretical-Concepts-Physics-Second-Alternative/dp/052152878X

Which then goes onto discuss how the attempts to model black body radiation broke down using classical physics thus paving the way for Planck and Einstein to introduce quantum mechanical ideas. It really is a fascinating story and shows that there is more to quantum mechanics than the development of Schrodinger's equation 

In general terms a good overall book on Thermodynamics is the classic by Zemansky 

https://www.amazon.co.uk/Heat-and-Thermodynamics-Fifth-edition/dp/B00X4VPZ0C/ref=sr_1_12?crid=32NLWLWHKDY3O&dchild=1&keywords=heat+and+thermodynamics+zemansky&qid=1633373359&s=books&sprefix=Zemansky+%2Cstripbooks%2C173&sr=1-12

Anyway thermodynamics has many applications Chandresekhar used it to work out the General equations of stellar structure without any need to know the internal structure of a star 

Here is my tribute to Thermodynamics and I would encourage people to study it in it's own right 

                                 Thermodynamics

 

Three laws oh so neat,

Describing the nature of heat.

The first says you cannot win,

You wont get back more than you put in.

 

But if it’s heat, there’s a permanent loss,

That is only regained at greater cost.

Finally there will come a great big chill,

Where all that there is, will stand still.

Lasers

 I was challenged by a work colleague to see if I could write a poem about lasers. The result is given below. Just like supeconductivity lasers are another successful application of quantum mechanics which again no agonising about it's meaning will ever produce a laser Ironically given that Einstein rejected the later formulation of quantum mechanics it was Einstein who worked out the basic theory of spontaneous emission on which the laser is based in 1918  This is a purely statistical argument, which again might surprise people as Einstein is allegedly supposed to have claimed that God does not play dice. Well in the early days of quantum mechanics it was Einstein who used statistical arguments to work out the consequences of the photo-electric effect and applied statistical reasoning to work out the Heat capacity of solids. So the idea that Einstein didn't like statistical reasoning is just incorrect. Indeed in his final years when he surveyed his debates with Bohr, he actually made the statement

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

So the only way to make sense of quantum mechanics according to Einstein is to endorse a statistical interpretation.

Anyway lasers are extremely useful devices but of course in the wrong hands can be used as terrifying weapons. Also idiots shine laser pens in pilots eyes, these people should be forced to face the consequences of their actions. In the right hands of course lasers are a benefit to mankind and a testimony to the ingenuity of scientists all over the world. Here is my tribute to them.




                                     Lasers

 

 Purest light that shines so bright,

 All because a photon takes flight.

Channelled by some clever means,

 Into a set of very intense beams.

 

Once the stuff of science fiction,

 It’s now part of our jurisdiction.

The wise will put you to good use,

But we must guard against abuse

.

James Bond concerned that Goldfinger’s Laser might destroy his manhood. I’ll leave you to decide whether or not that would have been a good thing  😀

Super Conductivity

 One of the most amazing applications of the formalism of quantum mechanics was the explanation by Baarden, Cooper and Schrieffer (BCS) of the phenomenon of Superconductivity. At low temperatures roughly about 4K it was noticed by Onnes that some metals appeared to have a dramatic reduction in their resistance. Whilst classical models were developed describing this phenomenon it wasn't until the 1950's that a microscopic version of the theory was developed. At first sight given the Pauli exclusion principle the phenomenon would seem impossible as it implies many electrons are occupying the same state. Which electrons being Fermions was impossible. Cooper realised at low temperatures there was the possiblity of an interaction between the lattice of the solid and the electrons causing them to effectively pair off. At low temperatures this interaction would be stable as the lattice vibrations would be relatively low. If electrons pair off their total spin becomes zero and they now behave like bosons for which it is possible for many bosons to occupy the same state. Thus the resistance of the metal is lowered BCS quickly realised that the properties of superconductors could be explained and they were awarded the Nobel prize for this work in 1972. 

Compared to the endless debates about the meaning or not of quantum mechanics which are going nowhere. This gives us a real insight into how nature works and ia a triumph of Mankinds ability to understand nature, something that will never come from discussing the meaning of the wave-function. As an interesting foot-note it was discovered in 1986 that some cuprates exhibited Superconductivity at much Higher temperatures than the 'normal ones' As yet there is no convincing explanation for this phenomenon so if you want a Nobel prize get cracking 😅

Here is my poem 

Super Conductivity

 

When it becomes very cold,

Nature becomes extremely bold.

All resistance suddenly dies,

An electric current really flies.

 

Electrons interact with the grid,

Combined in pairs they are hid.

This really ingenious tactic,

Was explained by a quantum mechanic¹⁾

1)    Someone who uses their knowledge of quantum mechanics to explain a feature of nature. In this case it was Cooper (what a clever fellow 😅😅 ).

 

  

Quantum Debates part II

 Ok I realise in the last post some people will say well what about the collapse of the wavefunction and the violation of the Bell inqualities don't they show that quantum mechanics is weird. I have covered this in many posts before but to save you looking here is a recap 

Let's take the so called collapse of the wavefunction first. This is usually illustrated by the so called cat paradox. If a quantum system can occupy a number of possible states then prior to measurement the sysrem is said to be in a superposition of states and on the usual story this means that when a measurement is made the so called wave function of the system collapses into one of the possible states with a given probability. Those who see the so called wave function as something more than the square root of a probability density function then make the leap to say that the measurement has caused the system to change from it's superposition to a single state. So there is something special about measurement and indeed quantum mechanics via this so called collapse vindicates idealism. So that we create reality by acts of measurement something which is alleged to be part of the Copenhagen interpretation which in fact it isn't. This should really be called the 'California Interpretation' and if it were true would indeed be weird and mysterious and all those who like to link Buddhism or Transcendental meditation with quantum mechanics would be vindicated. 

But this is totally unnecessary, lets take the classical situation for any statistical event if I know the underlying probability density function which summarises all the possiblities and I pick a given sample then I can calculate the probability that that person has a given height or a particular number will turn up if I throw a dice or spin a roulette wheel. Prior to the outcome I did not know what the outcome would be after the outcome I do. Thus the probability density function has 'collapsed' to give the particular outcome but all that is saying is that for classical systems the probability density function summarises all the possible outcomes and after the event has occured the outcome was realised with the given probability. I would argue the same is true of quantum mechanics before a measurement is made I do not know what the result would be but only the total outcomes with a given probability which if I know the appropriate solution to Schrodinger's wave equation I can calculate via the Born rule. Thus the superposition is not a real superposition but just a summary of possible outcomes of a measurement with the appropriate outcomes. The cat is definitely alive or dead before I open the box all I have done is updated my knowledge of the situation before hand. Opening the box hasn't caused the cat to be alive or dead but whether the radioactive poison was released or not. Something you could estimate if you know the half life of the radioactive material. All of this is consistent with quantum mechanics and how it is applied to calculate the probabilities of certain things happening and there is no need to invoke the collapse of the wavefunction to explain this process. 

In contrast to other realist interpretations such as those invoking hidden variables or the Many Worlds interpretaton nothing is added to the formalism. There is no need to invoke many worlds in a desperate attempt to maintain realism. My solution is robustly realist because the entities to which quantum mechanics is applied electrons, atoms and all the various exotic particles and there interactions are seen as real. Once one accepts that the so called wavefunction isn't anything physical but related to the probability density function via the Born rule, then there is no need to agonise whether or not the wave function is a physical object defined in (3N+1)*S space-time-spin dimensions for an N body system it makes no sense as a physical object but as a probability density function it makes a lot of sense. So by sticking to the statistical interpretation of Born then we can retain a fairly robust realism about the entities to which quantum mechanics is applied to and there is no need to worry about any form of idealism or mysticism. 

I'll talk about the Bell inequalities in a later post. But I defy anyone who disagrees with my perspective to show that my position is inconsistent with quantum mechanics. One can never observe a physical superposition of states because by definition any observation would collapse the wavefunction into one of the possible eigenstates. Those who claim otherwise are simply collating the reuslts of various measurements and claiming that this represents a real superposition. 

 

Quantum Debates

 Ok this poem is a rant against the endless debates about the alleged meaning of quantum mechanics. A debate I regard as totally pointless as the main issues were all settled by the following two rules of interpretation

1) The Borm rule which says that the modulus squared of the solution to Schrodinger's equation, when suitably normalised gives rise to a probability density function from which wc can calculate the expectation value of any variable. Conversely this means that the solution to Schrodinger's equation is effectively the square root of a probability density function not a real wave.

2) The link between the eigenvalues of the solution to Schrodinger's equation and the energy levels of the system under consideration. 

These rules are readily extendible to relativistic equations 

That really is all you need. plus the ability to use Schrodinger's equation or it's relativistic generalisation to understand the results of experiments and from which all the great successes of quantum mechanics have been applied to and which help us understand how nature works. 

 The rest whether or not the solution to Schrodinger's equation is a real wave in 3N+1 dimensional space time or merely a mathematical object which enables the probability that certain events will happen to be calculated or the energy levels of a system to be obtained. Whether there are multiple universess and so forth have nothing to do with physics at all. To coin a phrase quantum mechanics isn't mystical it is just statistical. 

Once the implications of this are accepted then much of the so called debate just dissolves. If there are no hidden variables then we can only have a statistical solution. But this does not mean that particles or their interactions are not real. just that their so called wave functions are not real, but only as real as say the Gaussian Distribution describing the distribution of heights of a collection of human beings in a sufficiently large enough sample.

 Instead of worrying about the meaining of the wave function try and apply the well trodden path of discovering the correct Hamiltonian to describe particles and how they interact with each other once this has been obtained just get on and calculate the consequences. If your Hamiltonian agrees with experimental results, then brilliant you have understood a part of nature. No amount of agonising about whether or not the solution to Schrodinger's equation is a real wave,. or signals travel faster than the speed of life is going to add one iota to the new knowledge. So let's celebrate all those physicists and chemists who daily apply quantum mechanics to understanding the world around us. Rather than the parasitic journalists and philosophers and I am afraid some prominent scientists such as Jim Al Khalili who jump on the band wagon of claiming that quantum mechanics is difficult to understand or mysterious. 

If you want to understand how quantum mechanics works, learn how to solve Schrodinger's equation or any of the relativistic equations don't waste your time reading what is essentially gibberish. Anyway here is my rant 

Quantum Debates

 

                                                  Many thoughts about you abound,

                                            Some of them seeming quite profound.

                                            But it’s all an illusion,

                                            Just adding to our confusion.

 

                                            It’s not mystical,

                                            It’s just statistical.

                                            Time to end this silly debate,

                                            And get on and calculate. 

Example of a silly debate about the meaning of quantum mechanics, that is going nowhere IMHO and there are plenty more around unfortunately.